## A location invariant moment-type estimator. I.(English)Zbl 1193.62052

Teor. Jmovirn. Mat. Stat. 76, 22-30 (2007) and Theory Probab. Math. Stat. 76, 23-31 (2008).
Let $$X_1,X_2,\dots,X_n$$ be i.i.d. random variables with common distribution function (d.f.) $$F(x)$$ and let $$X_{1,n},X_{2,n},\dots,X_{n,n}$$ be the associated order statistics. If there exist some numbers $$a_n>0$$, $$b_n\in\mathbb R$$ and some non-degenerate distribution $$G(x)$$ such that $P\{X_{n,n}\leq a_n x+b_n\}=F^n(x)\to G(x),\;n\to\infty,$ then $$G(x)$$ is equivalent to $G_{\gamma}(x)=\exp\{-(1+\gamma x)^{-1/\gamma}\},\;1+\gamma x>0,\;\gamma\not=0, \text{or}\;G_{\gamma}(x)=\exp\{-\exp(-x)\},\;x\in\mathbb R,\;\gamma=0.$ In this case $$F(x)$$ belongs to the domain of attraction of an extreme value d.f. $$G_{\gamma}$$ and $$\gamma$$ is referred to as the extreme value index.
In the last two decades many estimators of the extreme value index $$\gamma \in\mathbb R$$ are proposed that use upper order statistics. A.L.M. Dekkers, J.H.J. Einmahl and L. De Haan [Ann. Stat. 17, No. 4, 1833–1855 (1989; Zbl 0701.62029)] proposed the moment-type estimators $\hat{\gamma}_{n}^{M}(k)= M_{n}^{(1)} +1-2^{-1}\left\{ 1-\frac{(M_{n}^{(1)})^2}{M_{n}^{(2)}} \right\}^{-1},\;M_{n}^{(j)}=k^{-1}\sum_{i=0}^{k-1}\left(\log\frac{X_{n-i,n}}{X_{n-k,n}}\right)^j,\;j=1,2.$ The above estimators are scale invariant but not location invariant. M.I. Fraga Alves [Extremes 4, No. 3, 199–217 (2001; Zbl 1053.62063)] proposed a location invariant Hill type estimator given by $\hat{\gamma}_{n}^{H}(k_0,k)=k_0^{-1}\sum_{i=0}^{k_0-1}\log \left( \frac{X_{n-i,n}-X_{n-k,n}}{ X_{n-k_0,n}-X_{n-k,n}}\right),$ where $$k\to\infty\;k_0\to\infty\;,k/n\to0,\;k_0/k\to 0$$, and discussed its weak consistency, asymptotic expansion and the optimal choice of the sample fraction $$k_0$$.
In this paper, a general estimator for $$\gamma \in\mathbb R$$ based on the invariant Hill type estimator and the moment-type estimator is proposed. It is given by $\hat{\gamma}_{n}^{M}(k_0,k)= M_{n}^{(1)}(k_0,k) +1-2^{-1}\left\{ 1-\frac{(M_{n}^{(1)}(k_0,k))^2}{M_{n}^{(2)}(k_0,k)} \right\}^{-1},$
$M_{n}^{(j)}(k_0,k)=k_0^{-1}\sum_{i=0}^{k_0-1}\left(\log \frac{X_{n-i,n}-X_{n-k,n}}{ X_{n-k_0,n}-X_{n-k,n}} \right)^j,\;j=1,2.$ The weak and strong consistency of this new estimator are derived.

### MSC:

 62G05 Nonparametric estimation 62G32 Statistics of extreme values; tail inference 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference

### Citations:

Zbl 0701.62029; Zbl 1053.62063
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